Back in the days of film cameras, pro photographers carried a vast number of filters to deal with certain lighting conditions and to add effects. But, with the advent of DSLRs and their features such as white balance, many of these filters have now become obsolete.
However, some filters remain very useful in digital photography. The most popular filters are screw-on filters, which fit on to the front of DSLR camera lenses. These tend to be reasonably priced, but you will need to buy filters for each lens' thread size, which is listed in millimeters and can be found either on the front of the lens or on the back of the lens cap. Lens thread sizes range from about 48mm to 82mm on DSLRs.
The other thing to bear in mind is that any wide-angle lenses will require ultra-slim filters, which cuts down on the risk of vignetting at the edges of the photograph.
Fortunately, with the advent of DSLRs, there are far fewer essential filters to carry, but here are the ones I would still always keep with me.
While UV sunlight radiation doesn't create as many problems with DSLRs as it does with film cameras, the sunlight radiation can still cast a blueish hue over images. A UV filter can correct this problem without reducing the amount of visible light that reaches the image sensor.
However, the main reason to use a UV filter on all of your lenses is to protect them from dirt, dust, and -- most importantly -- accidental damage. If you're unlucky enough to drop a lens and it smashes, you'll be looking at hundreds of dollars worth of damage. But UV filters start from around $22 (Compare Prices), so the replacement cost will be a lot more reasonable! Buy a multicoated UV filter, otherwise you'll run the risk of lens flare with DSLRs. If I could only afford one filter, this would be it.
If you're interested in landscape photography, a polarizing filter is a must. Put simply, a polarizer reduces the amount of reflected light that goes to your camera's sensor. Blue skies appear a deeper blue, and reflections from water can be removed entirely. You can chose the amount of polarization that you add by twisting the outer ring of the filter, because this filter has two rings, one that attaches to the camera lens, and a free-form outer ring that twists for polarization. This adds polarization in degrees up to 180 degrees.
The downside of polarizing filters is that they greatly reduce the amount of light reaching the camera's sensor, often by two or three f-stops.
One last important point to note: Don't be tempted to buy the cheaper option of a "linear polarizer." These won't work with cameras that have autofocus or use TTL metering (Through The Lens) ... something that all DSLRs have.
Neutral Density Filter
The only purpose of a Neutral Density (ND) filter is to reduce the amount of light that reaches the camera's sensor. This can be particularly useful when a sufficiently long exposure isn't possible within the aperture parameters. An ND filter is most commonly used when photographing running water, because it helps create a smooth and etherial image. The ND filter can also be used to convey motion by adding blur to moving subjects and to make moving objects, such as cars, less obvious in landscape shots.
The most popular ND filters reduce light by two (ND4x or 0.6), three (ND8x or 0.9), or four (ND16x or 1.2) f-stops. It's unlikely that you'll find much use for more reduction than this, although some manufacturers make ND filters that reduce light by as many as six f-stops.
Graduated Neutral Density Filter
Graduated Neutral Density (GND), or Split, filters are an optional extra, but one that can prove useful if you don't like to do a lot of post-production work. These filters reduce the light at the top of the image and then smoothly graduate through to allowing a normal amount of light to hit the camera sensor from the lower portion of the image. These filters allow for capturing landscapes with very dramatic lighting, allowing both the sky and foreground to be correctly exposed.
How quickly the graduation and blend occurs depends on whether the filter is "soft" or "hard" edged, and this feature varies greatly from manufacturer to manufacturer. You need to do your research before buying these filters by looking at examples on manufacturers' websites. Like ND filters, GNDs are available in a variety of f-stop settings. You shouldn't need more than a one-to-three f-stop blend.